Amplitude model:

a_{ijk}= c_{k}f(m_{b}^{i}) s_{jk}x_{ij}^{-bk}e^{-akxij},

Take logarithms, correct for source scaling and discretize attenuation:i, j, k represent source, site and phase type

c_{k}source generation

f source scaling (taken as known function of m_{b})

s_{j}site effect

x_{ij}distance

b_{k}spreading

a_{k}spatial attenuation

D_{ijk}= C_{k}+ S_{jk}- b_{k}log_{10}x_{ij }- log_{10}e S_{m}a_{km}dx_{ijm},

Form phase ratios:m represents discretization

DDWrite in matrix form:_{ij}= DC + DS_{j}- Dblog_{10}x_{ij}- log_{10}e S_{m}Da_{m}dx_{ijm}.

Ax = dSolve using least squares:

x = (A^{t}W^{t}WA + S^{t}L^{2}S)^{-1}A^{t}W^{t}d,

S represents regularization equations

L regularization weights

W 1/data error

Calculate resolution, covariance and prediction error using standard formulae.

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